Related papers: On a function of Marcel Riesz
We explore some integrals associated with the Riesz function and establish relations to other functions from number theory that have appeared in the literature. We also comment on properties of these functions.
A new method for continuing the usual Dirichlet series that defines the Riemann zeta function ${\zeta}(s)$ is presented. Numerical experiments demonstrating the computational efficacy of the resulting continuation are discussed.
The behaviour of the generalised Riesz function defined by \[S_{m,p}(x)=\sum_{k=0}^\infty \frac{(-)^{k-1}x^k}{k! \zeta(mk+p)}\qquad (m\geq 1,\ p\geq 1)\] is considered for large positive values of $x$. A numerical scheme is given to compute…
We investigate the relation between the Riesz and the B{\'a}ez-Duarte criterion for the Riemann Hypothesis. In particular we present the relation between the function $R(x)$ appearing in the Riesz criterion and the sequence $c_k$ appearing…
The Riemann theta function is a complex-valued function of g complex variables. It appears in the construction of many (quasi-) periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation…
We investigate the relation between the Riesz and the Baez-Duarte criterion for the Riemann Hypothesis. In particular we present the relation between the function $R(x)$ appearing in the Riesz criterion and the sequence $c_k$ appearing in…
We note that the necessary and sufficient conditions established by Marcel Riesz for the inclusion of regular N\"orlund summation methods are in fact applicable quite generally.
The original criteria of Riesz and of Hardy-Littlewood concerning the truth of the Riemann Hypothesis (RH) are revisited and further investigated in light of the recent formulations and results of Maslanka and of Baez-Duarte concerning a…
A proof for the original Riemann hypothesis is proposed based on the infinite Hadamard product representation for the Riemann zeta function and later generalized to Dirichlet L-functions. The extension of the hypothesis to other functions…
In the first part we present the number theoretical properties of the Riemann zeta function and formulate the Riemann Hypothesis. In the second part we review some physical problems related to this hypothesis: the links with Random Matrix…
Franel and Landau derived an arithmetic statement involving the Farey sequence that is equivalent to the Riemann hypothesis. Since there is a relationship between the Mertens function and the Riemann hypothesis, there should be a…
This article derives the first two moments of the two versions of the Riesz distribution in the terms of their characteristic functions.
An integral formula is developed which applies to an essentially arbitrary function. An application is made to the Riemann zeta function.
We show that the higher derivatives of the Riemann zeta function may be expressed in terms of integrals involving the digamma function. Related integrals for the Stieltjes constants are also shown. We also present a formula for the…
This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation…
New recursion relations for the Riemann zeta function are introduced. Their derivation started from the standard functional equation. The new functional equations have both real and imaginary increment versions and can be applied over the…
This review article brings forth some recent results in the theory of the Riemann zeta-function $qzeta(s)$.
This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that…
The Hirsch function of a given continuous function is a new function depending on a parameter. It exists provided some assumptions are satisfied. If this parameter takes the value one, we obtain the well-known h-index. We prove some…
Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin…